Binysh, Jack and Alexander, Gareth P. (2018) Maxwell's theory of solid angle and the construction of knotted fields. Journal of Physics A: Mathematical and Theoretical, 51 (38). 385202. doi:10.1088/1751-8121/aad8c6

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Abstract

We provide a systematic description of the solid angle function as a means of constructing a knotted field for any curve or link in R3. This is a purely geometric construction in which all of the properties of the entire knotted field derive from the geometry of the curve, and from projective and spherical geometry. We emphasise a fundamental homotopy formula as unifying different formulae for computing the solid angle. The solid angle induces a natural framing of the curve, which we show is related to its writhe and use to characterise the local structure in a neighbourhood of the knot. Finally, we discuss computational implementation of the formulae derived, with C code provided, and give illustrations for how the solid angle may be used to give explicit constructions of knotted scroll waves in excitable media and knotted director fields around disclination lines in nematic liquid crystals.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Divisions:	Faculty of Science > Centre for Complexity Science Faculty of Science > Mathematics Faculty of Science > Physics
Library of Congress Subject Headings (LCSH):	Knot theory, Geometry
Journal or Publication Title:	Journal of Physics A: Mathematical and Theoretical
Publisher:	Institute of Physics
Official Date:	21 August 2018
Dates:	Date Event 21 August 2018 Published 8 August 2018 Accepted 25 May 2018 Submitted
Date of first compliant deposit:	21 August 2018
Volume:	51
Number:	38
Article Number:	385202
DOI:	10.1088/1751-8121/aad8c6
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
RIOXX Funder/Project Grant:	Project/Grant ID RIOXX Funder Name Funder ID EP/L015374/1 [EPSRC] Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266 EP/N007883/1 [EPSRC] Engineering and Physical Sciences Research Council http://dx.doi.org/10.13039/501100000266 David Crighton Fellowship University of Cambridge. Department of Applied Mathematics and Theoretical Physics http://viaf.org/viaf/157726551
Open Access Version:	Publisher

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